ARTICLE IN PRESS JID: JOEMS [m5G;February 10, 2017;22:55] Journal of the Egyptian Mathematical Society 0 (2017) 1–7 Contents lists available at ScienceDirect Journal of the Egyptian Mathematical Society journal homepage: www.elsevier.com/locate/joems Original Article Fuzzy soft connected sets in fuzzy soft topological spaces II A Kandil a, O.A El-Tantawy b, S.A El-Sheikh c, Sawsan S.S El-Sayed c,∗ a Mathematics Department, Faculty of Science, Helwan University, Helwan, Egypt Mathematics Department, Faculty of Science, Zagazig University, Zagazig, Egypt c Mathematics Department, Faculty of Education, Ain Shams University, Cairo, Egypt b a r t i c l e i n f o Article history: Received 18 October 2016 Revised 27 December 2016 Accepted January 2017 Available online xxx Keywords: Fuzzy soft Fuzzy soft Fuzzy soft Fuzzy soft Fuzzy soft a b s t r a c t In this paper, we introduce some different types of fuzzy soft connected components related to the different types of fuzzy soft connectedness and based on an equivalence relation defined on the set of fuzzy soft points of X We have investigated some very interesting properties for fuzzy soft connected components We show that the fuzzy soft C5 -connected component may be not exists and if it exists, it may not be fuzzy soft closed set Also, we introduced some very interesting properties for fuzzy soft connected components in discrete fuzzy soft topological spaces which is a departure from the general topology sets topological space separated sets connected sets connected components Introduction The concept of a fuzzy set was introduced by Zadeh [15] in his classical paper of 1965 In 1968, Chang [2] gave the definition of fuzzy topology Since Chang applied fuzzy set theory into topology many topological notions were investigated in a fuzzy setting In 1999, the Russian researcher Molodtsov [9] introduced the soft set theory which is a completely new approach for modeling uncertainty He established the fundamental results of this new theory and successfully applied the soft set theory into several directions Maji et al [8] defined and studied several basic notions of soft set theory in 2003 Shabir and Naz [12] introduced the concept of soft topological space Maji et al [7] initiated the study involving both fuzzy sets and soft sets In this paper, Maji et al combined fuzzy sets and soft sets and introduced the concept of fuzzy soft sets In 2011, Tanay Kandemir [14] gave the topological structure of fuzzy soft sets The notions of fuzzy soft connected sets and fuzzy soft connected components are very important in fuzzy soft topological spaces which in turn reflect the intrinsic nature of it that is in fact its peculiarity In fuzzy soft setting, connectedness has been introduced by Mahanta and Das [6] and Karatas¸ et al [5] Recently, Kandil et al [4] introduced some types of separated sets and some types of connected sets They studied the relationship between these types ∗ Corresponding author E-mail addresses: sawsan_809@yahoo.com, s.elsayed@mu.edu.sa (S.S.S El-Sayed) © 2017 Egyptian Mathematical Society Production and hosting by Elsevier B.V This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) In this paper, we extend the notion of connected components of fuzzy topological space to fuzzy soft topological space In Section 3, we introduce and investigate some very interesting properties for fuzzy soft connected components We define an equivalence relation on the set of fuzzy soft points The union of equivalence classes turns out to be a maximal fuzzy soft connected set which is called a fuzzy soft connected component There are many types of connected components deduced from the many types of connected sets due to Kandil et al [4] Furthermore, we show that some of these connected components may be not exists and the some if exists, it may not be fuzzy soft closed set Moreover, we introduced some very interesting properties for fuzzy soft connected components in discrete fuzzy soft topological spaces which is a departure from the general topology Preliminaries Throughout this paper X denotes initial universe, E denotes the set of all possible parameters which are attributes, characteristic or properties of the objects in X In this section, we present the basic definitions and results of fuzzy soft set theory which will be needed in the sequel Definition 2.1 [2] A fuzzy set A of a non-empty set X is characterized by a membership function μA : X −→ [0, 1] = I whose value μA (x) represents the “degree of membership” of x in A for x ∈ X Let IX denotes the family of all fuzzy sets on X http://dx.doi.org/10.1016/j.joems.2017.01.006 1110-256X/© 2017 Egyptian Mathematical Society Production and hosting by Elsevier B.V This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Please cite this article as: A Kandil et al., Fuzzy soft connected sets in fuzzy soft topological spaces II, Journal of the Egyptian Mathematical Society (2017), http://dx.doi.org/10.1016/j.joems.2017.01.006 ARTICLE IN PRESS JID: JOEMS [m5G;February 10, 2017;22:55] A Kandil et al / Journal of the Egyptian Mathematical Society 000 (2017) 1–7 Definition 2.2 [9] Let A be a non-empty subset of E A pair (F, A) denoted by FA is called a soft set over X , where F is a mapping given by F: A → P(X) In other words, a soft set over X is a parametrized family of subsets of the universe X For a particular e ∈ A , F(e) may be considered the set of e-approximate elements of the soft set (F, A) and if e ∈ A, then F (e ) = φ i.e F = {F (e ) : e ∈ A ⊆ E, F : A → P (X )} Definition 2.13 [10,11] Let (X, τ , E) be a fuzzy soft topological space and fA ∈ FSS(X)E The fuzzy soft closure of fA , denoted by Fcl(fA ) is the intersection of all fuzzy soft closed supersets of fA , i.e F cl ( fA ) = {hC ; hC ∈ τ c and fA ⊆ hC } Clearly, Fcl(fA ) is the smallest fuzzy soft closed set over X which contains fA , and Fcl(fA ) is fuzzy soft closed set Aktas¸ and Çag˘ man [1] showed that every fuzzy set may be considered as a soft set That is, fuzzy sets are a special class of soft sets Definition 2.14 [11,13] The fuzzy soft set fA ∈ FSS(X)E is called fuzzy soft point if there exist x ∈ X and e ∈ E such that μef (x ) = α ; Definition 2.3 [7] Let A⊆E A pair (f, A), denoted by fA , is called fuzzy soft set over X , where f is a mapping given by f : A −→ IX defined by fA (e ) = μef ; where μef = if e ∈ A, and μef = if e ∈ A A A A, where 0(x ) = 0∀ x ∈ X The family of all these fuzzy soft sets over X denoted by FSS(X)E A (0 ≤ α ≤ 1) and μef (y ) = ∀y ∈ X − {x} and this fuzzy soft point A is denoted by xeα or fe The class of all fuzzy soft points of X, denoted by FSP(X)E Definition 2.15 [6] The fuzzy soft point xeα is said to be belonging to the fuzzy soft set fA , denoted by xeα ∈ fA , if for the element e ∈ A, α ≤ μef (x ) If xeα is not belong to fA , we write xeα ∈/ fA and implies A Definition 2.4 [3,7,10,11,13,14] The complement of a fuzzy soft set (f, A) , denoted by (f, A)c , and defined by (f, A)c = ( f c , A ) , fAc : A −→ IX is a mapping given by μef c = − μef ∀e ∈ A Clearly, ( fAc )c A = fA A Definition 2.5 [7,10,11,13,14] A fuzzy soft set fE over X is said to be a null-fuzzy soft set, denoted by 0E , if for all e ∈ E, fE (e ) = Definition 2.6 [7,10,11,13,14] A fuzzy soft set fE over X is said to be an absolute fuzzy soft set, denoted by 1E , if fE (e ) = ∀e ∈ E Clearly, we have (0E )c = 1E and (1E )c = 0E Definition 2.7 [3,7,10,11,13,14] Let fA , gB ∈ FSS(X)E Then fA is fuzzy soft subset of gB , denoted by fA ⊆ gB , if A⊆B and μef (x ) ≤ that α > μef (x ) A Definition 2.16 [11,13] A fuzzy soft point xeα is said to be a quasicoincident with a fuzzy soft set fA , denoted by xeα q fA , if α + μef (x ) > Otherwise, xeα is non-quasi-coincident with fA and deA noted by xeα q fA Definition 2.17 [11,13] A fuzzy soft set fA is said to be quasicoincident with gB , denoted by fA q gB , if there exists x ∈ X such that μef (x ) + μegB (x ) > 1, for some e ∈ A ∩ B If this is true we can A say that fA and gB are quasi-coincident at x Otherwise, fA and gB are not quasi-coincident and denoted by fA q gB μegB (x )∀x ∈ X, ∀e ∈ E Also, gB is called fuzzy soft superset of fA de- Proposition 2.1 [11, 13] Let fA and gB be two fuzzy soft sets Then, fA ⊆ gB if and only if fA q (gB )c In particular, xeα ∈ fA if and only if xeα q (fA )c Definition 2.8 [3,7,10,11,13,14] Two fuzzy soft sets fA and gB on X are called equal if fA ⊆ gB and gB ⊆ fA Definition 2.18 [10] Let FSS(X)E and FSS(Y)K be families of fuzzy soft sets over X and Y, respectively Let u : X −→ Y and p : E −→ K be mappings Then the map fpu is called fuzzy soft mapping from FSS(X)E to FSS(Y)K , denoted by fpu : FSS(X)E −→ FSS(Y)K , such that: A noted by gB ⊇ fA If fA is not fuzzy soft subset of gB , we written as fA gB Definition 2.9 [7,10,11,13,14] The union of two fuzzy soft sets fA and gB over the common universe X, denoted by fA ࣶgB , is also a fuzzy soft set hC , where C = A ∪ B and for all e ∈ C, hC (e ) = μeh = μef A ∨ μegB ∀e ∈ E C Definition 2.10 [7,10,11,13,14] The intersection of two fuzzy soft sets fA and gB over the common universe X, denoted by fA ࣵgB , is also a fuzzy soft set hC , where C = A ∩ B and for all e ∈ C, hC (e ) = μeh = μef ∧ μegB ∀e ∈ E C A Definition 2.11 [14] Let FSS(X)E be a collection of fuzzy soft sets over a universe X with a fixed set of parameters E Then τ ⊆FSS(X)E is called fuzzy soft topology on X if 0E , 1E ∈ τ , where 0E (e ) = and 1E (e ) = 1∀e ∈ E, The union of any members of τ belongs to τ The intersection of any two members of τ belongs to τ The triplet (X, τ , E) is called fuzzy soft topological space over X Also, each member of τ is called fuzzy soft open set in (X, τ , E) Definition 2.12 [14] Let (X, τ , E) be a fuzzy soft topological space A fuzzy soft set fA over X is said to be fuzzy soft closed set in X, if its relative complement fAc is fuzzy soft open set If gB ∈ FSS(X)E , then the image of gB under the fuzzy soft mapping fpu is a fuzzy soft set over Y defined by fpu (gB ) where ∀k ∈ p(E), ∀y ∈ Y, f pu (gB )(k )(y ) = ∨ [ ∨ (gB (e ))](x ) if x ∈ u−1 (y ),0 u(x )=y p(e )=k If hC ∈ FSS(Y)K , then the pre-image of hC under the fuzzy soft −1 mapping fpu , f pu (hC ) is a fuzzy soft set over X defined by ∀e ∈ −1 p ( K ), ∀x ∈ X, −1 f pu (hC )(e )(x ) = hC ( p(e ))(u(x )) for p(e ) ∈ C,0 Definition 2.19 [10] The fuzzy soft mapping fpu is called surjective (resp injective) if p and u are surjective (resp injective), also fpu is said to be constant if p and u are constant Definition 2.20 [10] Let (X, τ , E) and (Y, τ , K) be two fuzzy soft topological spaces and fpu : FSS(X)E −→ FSS(Y)K be a fuzzy soft mapping Then fpu is called: −1 Fuzzy soft continuous if f pu (hC ) ∈ τ1 ∀ hC ∈ τ Fuzzy soft open if fpu (gB ) ∈ τ ∀ gB ∈ τ Please cite this article as: A Kandil et al., Fuzzy soft connected sets in fuzzy soft topological spaces II, Journal of the Egyptian Mathematical Society (2017), http://dx.doi.org/10.1016/j.joems.2017.01.006 ARTICLE IN PRESS JID: JOEMS [m5G;February 10, 2017;22:55] A Kandil et al / Journal of the Egyptian Mathematical Society 000 (2017) 1–7 Definition 2.21 [5] Two non-null fuzzy soft sets fE and gE are said to be fuzzy soft Q-separated in a fuzzy soft topological space (X, τ , E) if Fcl(fE )ࣵ gE = fE F cl (gE ) = 0E Definition 2.22 [5] Let (X, τ , E) be a fuzzy soft topological space and fE ∈ FSS(X)E Then, fE is called: FSC1 -connected: if does not exist two non-null fuzzy soft open sets hE and sE such that fE ⊆ hE ࣶsE , hE ࣵsE ⊆ fEc , fE hE = 0E , and f E sE = 0E FSC2 -connected: if does not exist two non-null fuzzy soft open sets hE and sE such that fE ⊆ hE ࣶsE , fE hE sE = 0E , fE hE = 0E , and fE sE = 0E FSC3 -connected: if does not exist two non-null fuzzy soft open sets hE and sE such that fE ⊆ hE ࣶsE , hE ࣵsE ⊆ fEc , hE fEc , and sE c fE FSC4 -connected: if does not exist two non-null fuzzy soft open sets hE and sE such that fE ⊆ hE ࣶsE , fE hE sE = 0E , hE fEc , and fEc Otherwise, fE is called FSCi -disconnected set for i = 1, 2, 3, In the above definition, if we take 1E instead of fE , then the fuzzy soft topological space (X, τ , E) is called FSCi -connected space ( i = 1, 2, 3, ) sE FSCM -disconnected set if there exist two non-null fuzzy soft Qseparated sets hE , sE in X such that fE = hE sE Otherwise, fE is called FSCM -connected set FSCS -disconnected set if there exist two non-null fuzzy soft weakly-separated sets hE , sE in X such that fE = hE sE Otherwise, fE is called FSCS -connected set FSO-disconnected (respectively, FSOq -disconnected) set if there exist two non-null fuzzy soft separated (respectively, strongly separated) sets hE , sE in X such that fE = hE sE Otherwise, fE is called FSO-connected (respectively, FSOq -connected) set FSC5 -connected set in X if there does not exist any non-null proper fuzzy soft clopen set in (fE , τ fE , E) Note that, this kind of fuzzy soft connectedness was studied by Mahanta and Das [6], Shabir and Naz [12] In the above definitions, if we take 1E instead of fE , then the fuzzy soft topological space (X, τ , E) is called FSCM -connected (respectively, FSCS -connected, FSO-connected, FSOq -connected, FSC5 connected) space Remark 2.3 [4] In a fuzzy soft topological space (X, τ , E) The classes of FSO-connected, FSOq -connected, and FSCi -connected sets for i = 1, 2, 3, 4, S, M can be described by the following diagram Remark 2.1 [5] The relationship between FSCi -connectedness (i = 1, 2, 3, ) can be described by the following diagram: F SC1 ⇓ F SC ⇒ ⇒ F SC F SC ↓ F SC S F SC ⇓ F SC ↓ F SC F SC M Definition 2.24 [4] Let fE ∈ FSS(X)E The support of fE (e), denoted by S(fE (e)), is the set, S( fE (e )) = {x ∈ X;fE (e)(x) > 0} Definition 2.25 [4] Two fuzzy soft sets fE and gE are said to be quasi-coincident with respect to fE if μef (x ) + μegE (x ) > for every E x ∈ S(fE (e)) ↓ F SOq Equivalence relations and components In disconnected fuzzy soft topological space (X, τ , E), the universe fuzzy soft set 1E can be decomposed into several pieces of fuzzy soft sets, each of which is connected As in general topological space, the whole space is decomposed into components In fuzzy soft setting, this decomposition is obtained in form of unions of equivalence classes of a certain equivalence relation, defined on the set of fuzzy soft points in X The union of equivalence classes turns out to be a maximal fuzzy soft connected set Accordingly, we have many types of notions of components in fuzzy soft setting e Definition 2.26 [4] Two non-null fuzzy soft sets fE and gE are said to be fuzzy soft strongly separated in a fuzzy soft topological space (X, τ , E) if there exist hE and sE ∈ τ such that fE ⊆ hE , gE ⊆ sE , fE sE = gE hE = 0E , fE , hE are fuzzy soft quasi-coincident with respect to fE , and gE , sE are fuzzy soft quasi-coincident with respect to gE Remark 2.2 [4] In fuzzy soft topological space (X, τ , E) the relationship between different notions of fuzzy soft separated sets can be described by the following diagram fuzzy soft strongly separated ⇓ fuzzy soft separated fuzzy soft Q -separated ←→ F SO F SC Definition 2.23 [4] Two non-null fuzzy soft sets fE and gE are said to be: Weakly separated sets in a fuzzy soft topological space (X, τ , E) if Fcl(fE ) q gE and fE q Fcl(gE ) Separated sets in a fuzzy soft topological space (X, τ , E) if there exist non-null fuzzy soft open sets hE and sE such that fE ⊆ hE , gE ⊆ sE and fE sE = gE hE = 0E ⇒ ⇓ fuzzy soft weakly separated e e connected set fA such that xα1 ∈ fA and yβ2 ∈ fA for i = , 2, S, M, O, Oq } Then, Ei is an equivalence relation on FSP(X)E Proof As a sample we will prove the case of i = Reflexivity follows from the fact that for each fuzzy soft point xeα in X, there exists a fuzzy soft point xe1 in X, which is a FSC1 -connected and obviously contains xeα Symmetry is obvious To show transitivity, let e e e e e xα1 , yβ2 and zγ3 be fuzzy soft points in X such that (xα1 , yβ2 ) ∈ E1 e e and (yβ2 , zγ3 ) ∈ E1 Then, there exist FSC1 -connected sets fA and gB e e e e in X such that xα1 ∈fA , yβ2 ∈ fA and yβ2 ∈ gB , zγ3 ∈ gB Therefore, β ≤ μef2 (y ) and β ≤ μeg2B (y ) Hence, fA gB = 0E So by Theorem A Definition 2.27 [4] A fuzzy soft set fE in a fuzzy soft topological space (X, τ , E) is called: e Proposition 3.1 For fuzzy soft points xα1 and yβ2 in X define a relation Ei as follows: e e e e Ei = {(xα1 , yβ2 ); xα1 , yβ2 ∈ FSP(X)E and there exists a FSCi - e 4.10 in [9], fA ࣶgB is a FSC1 -connected Also, we have xα1 ∈ fA ࣶgB e and zγ3 ∈ fA ࣶgB Therefore, E1 is an equivalence relation Similarly, Ei is an equivalence relation for i = 2, S, M, O, Oq Please cite this article as: A Kandil et al., Fuzzy soft connected sets in fuzzy soft topological spaces II, Journal of the Egyptian Mathematical Society (2017), http://dx.doi.org/10.1016/j.joems.2017.01.006 ARTICLE IN PRESS JID: JOEMS [m5G;February 10, 2017;22:55] A Kandil et al / Journal of the Egyptian Mathematical Society 000 (2017) 1–7 Let xeα be a fuzzy soft point in X and Ei be the equivalence relation on FSP(X)E , described as above Then the equivalence class determined by xeα , is denoted by Ei (xeα ) for i = 1, 2, S, M, O, Oq Definition 3.1 The union ࣶ Ei (xeα ) of all fuzzy soft points contained in the equivalence class Ei (xeα ) is called a Ci -component of the universe fuzzy soft set 1E , determined by xeα We denoted it by Ci (xeα ) for i = 1, 2, S, M, O, Oq Theorem 3.1 For each fuzzy soft point xeα ∈ FSP(X)E , the component Ci (xeα ) is the maximal FSCi -connected (respectively, FSO-connected, FSOq -connected) set in X containing xeα for i = 1, 2, S, M Proof As a sample we will prove the case of C1 (xeα ) Let xeα ∈ FSP(X)E and let {(fA )i ; i ∈ I} be the family of FSC1 -connected sets in X, containing xeα We claim C1 (xeα ) = ( fA )i i∈I Firstly, we show that μe ( f A )i ( y ) = i∈I ( fA )i ⊆ C1 ( xe α ) Let y ∈ X, e ∈ E, βi for each i ∈ I and supβi = β Then, μe i∈I supβi = β i∈I ( f A )i ( y ) = i∈I Now, if β = 0, we have nothing to prove Suppose β = Then for every real number > 0, there exists i ∈ I such that μe( f ) (y ) = A i βi > β − Therefore, for each fuzzy soft point yeβ − where < < β , there exists a fuzzy soft set (fA )i such that yeβ − ∈ (fA )i Since (fA )i is a FSC1 -connected set, containing xeα , it follows that (xeα , yeβ − ) ∈ E1 and hence yeβ − ∈ E1 (xeα ) for every < < β Now, let {(yeβ ) j ; j ∈ J } be the family of all fuzzy soft points in X with support y which are E1 -related to xeα Then {yeβ − }0<